Gradient pattern analysis: Difference between revisions

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'''Gradient Pattern Analysis''' (GPA)<ref name=rosa2000>Rosa, R.R., Pontes, J., PontesChristov, C. I., ChristovRamos, F. M., RamosRodrigues Neto, C., Rodrigues NetoRempel, E. L., RempelWalgraef, D. Walgraef, ''Physica A'' '''283''', 156 (2000).</ref> is a geometric computing method for characterizing [[symmetry breaking]] of an ensemble of asymmetric vectors regularly distributed in a square lattice. Usually, the lattice of vectors represent the first-order gradient of a scalar field, here an ''M x M'' square amplitude matrix. An important property of the gradient representation is the following: A given ''M x M'' matrix where all amplitudes are different results in an ''M x M'' gradient lattice containing <math>N_{V} = M^2</math> asymmetric vectors. As each vector can be characterized by its norm and phase, variations in the <math>M^2</math> amplitudes can modify the respective <math>M^2</math> gradient pattern.

The original concept of GPA was introduced by Rosa, Sharma and Valdivia in 1999<ref name=Rosa99>Rosa, R.R.; Sharma, A. S. Sharma and J. Valdivia, J.A. ''Int. J. Mod. Phys. C'', '''10''', 147 (1999).</ref> Usually GPA is applied for spatio-temporal pattern analysis in physics and environmental sciences operating on time-series and digital images.

== Calculation ==

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For a complex extended pattern (matrix of amplitudes of a spatio-temporal pattern) composed by locally asymmetric fluctuations, <math>G_{A}</math> is nonzero, defining different classes of irregular fluctuation patterns (1/f noise, chaotic, reactive-diffusive, etc).

Besides <math>G_{A}</math> other measurements (called ''gradient moments'') can be calculated formfrom the gradient lattice.<ref name=rosa03>Rosa, R.R.; Campos, M. R.; CamposRamos, F. M.; RamosVijaykumar, N. L.; VijaykumarFujiwara, S.; FujiwaraSato, T. Sato, ''Braz. JourJ. Phys.'' '''33''', 605 (2003).</ref>. Considering the sets of local norms and phases as discrete compact groups, spatially distributed in a square lattice, the gradient moments have the basic property of being globally invariant (for rotation and modulation).

== Relation to other methods ==

When GPA is conjugated with wavelet analysis, then the method is called ''Gradient Spectral Analysis'' (GSA), usually applied to short time series analysis.<ref name=rosa08>Rosa, R.R. et al., ''Advances in Space Research'' '''42''', 844 (2008), [doi:10.1016/j.asr.2007.08.015].</ref>

== References ==

{{reflist}}

<references/>

{{refbegin}}

* Assireu, A.T., R. R. Rosa, N. L. Vijaykumar, J. A. Lorenzetti, E. L. Rempel, F. M. Ramos, L. D. Abreu Sá, M. J. A. Bolzan, A. Zanandrea, ''Physica D'' '''68''', 397 (2002).

* Rosa, R.R., M. R. Campos, F. M. Ramos, N. L. Vijaykumar, S. Fujiwara, T. Sato, ''Braz. Jour. Phys.'' '''33''', 605 (2003).

* Baroni, M.P.M.A, R. R. Rosa, A. Ferreira da Silva, I. Pepe, L. S. Roman, F. M. Ramos, R. Ahuja, C. Persson, E. Veje, ''Microelectronics Journal'' '''37''', 290 (2006).

* Rosa, R.R., M. P. M. A. Baroni, G. T. Zaniboni, A. Ferreira da Silva, L. S. Roman, J. Pontes and M. J. A. Bolzan, ''Physica A'' '''386''', 666 (2007).

{{refend}}